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单词 HermitianFormOverADivisionRing
释义

Hermitian form over a division ring


Let D be a division ring admitting an involutionMathworldPlanetmath (http://planetmath.org/Involution2) *. Let V be a vector spaceMathworldPlanetmath over D. A Hermitian formMathworldPlanetmathPlanetmath over D is a function from V×V to D, denoted by (,) with the following properties, for any v,wV and dD:

  1. 1.

    (,) is additive in each of its arguments,

  2. 2.

    (du,v)=d(u,v),

  3. 3.

    (u,dv)=(u,v)d*,

  4. 4.

    (u,v)=(v,u)*.

Note that if the Hermitian form (,) is non-trivial and if * is the identity on D, then D is a field and (,) is just a symmetric bilinear formMathworldPlanetmath.

If we replace the last condition by (u,v)=-(v,u)*, then (,) over D is called a skew Hermitian form.

Remark. Every skew Hermitian form over a division ring induces a Hermitian form and vice versa.

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更新时间:2025/5/3 12:08:02